.. _oneapi-mkl-rng-cauchy:

oneapi::mkl::rng::cauchy
========================


Generates Cauchy distributed random values.


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Description
***********

The ``oneapi::mkl::rng::cauchy`` class object is used in the
``oneapi::mkl::rng::generate`` function to provide Cauchy distributed random numbers with displacement (``a``) and scalefactor (``b``, ``β``), where :math:`a, \beta \in R ; \beta > 0`.


The probability density function is given by:

.. math::

   F_{a, \beta}(x) =
   \frac{1}
   {\pi \beta
   \left(
      1 + {\left( \frac{x-a}{\beta} \right)}^2
   \right)},
   - \infty < x < + \infty


The cumulative distribution function is as follows:

.. math::

   F_{a, \beta}(x) =
   \frac{1}{2} +
   \frac{1}{\pi} \arctan (\frac{x-a}{\beta}),
   - \infty < x < + \infty



.. list-table::
   :header-rows: 1

   * - Product and Performance Information
   * - Performance varies by use, configuration and other factors. Learn more at `https://www.intel.com/PerformanceIndex <https://www.intel.com/PerformanceIndex>`__. Notice revision #20201201


API
***


Syntax
------

.. code-block:: cpp


    template<typename RealType = float,
      typename Method = cauchy_method::by_default>
      class cauchy {
      public:
      using method_type = Method;
      using result_type = RealType;
      cauchy(): cauchy((RealType)0.0, (RealType)1.0){}
      explicit cauchy(RealType a, RealType b);
      explicit cauchy(const param_type& pt);
      RealType a() const;
      RealType b() const;
      param_type param() const;
      void param(const param_type& pt);
    };


Devices supported: Host, CPU, and GPU.

Include Files
-------------

- ``oneapi/mkl/rng.hpp``


Template Parameters
-------------------

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   * - ``typename RealType = float``
     - Type of the produced values. The specific values are as follows:

       ``float``

       ``double``
   * - ``typename Method = oneapi::mkl::rng::cauchy_method::by_default``
     - Generation method. The specific values are as follows:
       ``oneapi::mkl::rng::cauchy_method::icdf``

       See brief descriptions of the methods in :ref:`distributions-template-parameter-method`


Input Parameters
----------------

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   * - Name
     - Type
     - Description
   * - a
     - ``RealType (float, double)``
     - Displacement ``a``.
   * - b
     - ``RealType (float, double)``
     - Scalefactor ``b``.